A rigorous dynamic process model should be developed to increase the understanding about the operation fundamentals and to test the control hypothesis.. [1] 1 Introduction to Process Mod
Trang 2Process Dynamics and Control
Modeling for Control and Prediction
Trang 4Process Dynamics and Control
Trang 6Process Dynamics and Control
Modeling for Control and Prediction
Trang 7The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England Telephone (+44) 1243 779777
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Library of Congress Cataloging-in-Publication Data
Roffel, Brian
Process dynamics and control : modeling for control and prediction / Brian
Roffel and Ben Betlem
Typeset by SNP Best-set Typesetter Ltd., Hong Kong
Printed and bound in Great Britain by TJ International Ltd, Padstow, Cornwall
This book is printed on acid-free paper responsibly manufactured from sustainable forestry in which at least two trees are planted for each one used for paper production
Trang 8CONTENTS
Foreword xi
Preface xiii
Acknowledgement xv
1 Introduction to Process Modeling 1
1.1 Application of Process Models 1
1.2 Dynamic Systems Modeling 2
1.3 Modeling Steps 5
1.4 Use of Diagrams 16
1.5 Types of Models 20
1.6 Continuous versus Discrete Models 23
References 23
2 Process Modeling Fundamentals 25
2.1 System States 25
2.2 Mass Relationship for Liquid and Gas 29
2.3 Energy Relationship 38
2.4 Composition Relationship 48
3 Extended Analysis of Modeling for Process Operation 57
3.1 Environmental Model 57
3.2 Procedure for the Development of an Environmental Model for Process Operation 58
3.3 Example: Mixer 68
3.4 Example: Evaporator with Variable Heat Exchanging Surface 69
4 Design for Process Modeling and Behavioral Models 71
4.1 Behavioral Model 71
4.2 Example: Mixer 77
5 Transformation Techniques 81
5.1 Introduction 81
5.2 Laplace Transform 81
5.3 Useful Properties of Laplace Transform: limit functions 83
5.4 Transfer Functions 84
5.5 Discrete Approximations 89
5.6 z-Transforms 90
References 95
6 Linearization of Model Equations 97
6.1 Introduction 97
6.2 Non-linear Process Models 97
6.3 Some General Linearization Rules 100
6.4 Linearization of Model of the Level Process 102
6.5 Linearization of the Evaporator model 103
6.6 Normalization of the Transfer Function 105
6.7 Linearization of the Chemical Reactor Model 105
Trang 97 Operating Points 109
7.1 Introduction 109
7.2 Stationary System and Operating Point 109
7.3 Flow Systems 110
7.4 Chemical System 111
7.5 Stability in the Operating Point 113
7.6 Operating Point Transition 116
8 Process Simulation 119
8.1 Using Matlab Simulink 119
8.2 Simulation of the Level Process 119
8.3 Simulation of the Chemical Reactor 124
References 126
9 Frequency Response Analysis 127
9.1 Introduction 127
9.2 Bode Diagrams 129
9.3 Bode Diagram of Simulink Models 135
References 137
10 General Process Behavior 139
10.1 Introduction 139
10.2 Accumulation Processes 140
10.3 Lumped Process with Non-interacting Balances 142
10.4 Lumped Process with Interacting Balances 144
10.5 Processes with Parallel Balances 148
10.6 Distributed Processes 151
10.7 Processes with Propagation Without Feedback 154
10.8 Processes with Propagation With Feedback 157
11 Analysis of a Mixing Process 161
11.1 The Process 161
11.2 Mixer with Self-adjusting Height 164
12 Dynamics of Chemical Stirred Tank Reactors 169
12.1 Introduction 169
12.2 Isothermal First-order Reaction 169
12.3 Equilibrium Reactions 172
12.4 Consecutive Reactions 175
12.5 Non-isothermal Reactions 178
13 Dynamic Analysis of Tubular Reactors 185
13.1 Introduction 185
13.2 First-order Reaction 186
13.3 Equilibrium Reaction 188
13.4 Consecutive Reactions 188
13.5 Tubular Reactor with Dispersion 188
13.6 Dynamics of Adiabatic Tubular Flow Reactors 192
References 194
Trang 10Contents [vii]
14 Dynamic Analysis of Heat Exchangers 195
14.1 Introduction 195
14.2 Heat Transfer from a Heating Coil 195
14.3 Shell and Tube Heat Exchanger with Condensing Steam 198
14.4 Dynamics of a Counter-current Heat Exchanger 205
References 206
15 Dynamics of Evaporators and Separators 207
15.1 Introduction 207
15.2 Model Description 208
15.3 Linearization and Laplace Transformation 209
15.4 Derivation of the Normalized Transfer Function 210
15.5 Response Analysis 211
15.6 General Behavior 212
15.7 Example of Some Responses 212
15.8 Separation of Multi-phase Systems 213
15.9 Separator Model 214
15.10 Model Analysis 215
15.11 Derivation of the Transfer Function 217
16 Dynamic Modeling of Distillation Columns 219
16.1 Column Environmental Model 219
16.2 Assumptions and Simplifications 220
16.3 Column Behavioral Model 221
16.4 Component Balances and Equilibria 222
16.5 Energy Balances 225
16.6 Tray Hydraulics 228
16.7 Tray Pressure Drop 233
16.8 Column Dynamics 236
Notation 240
Greek Symbols 242
References 243
17 Dynamic Analysis of Fermentation Reactors 245
17.1 Introduction 245
17.2 Kinetic Equations 245
17.3 Reactor Models 247
17.4 Dynamics of the Fed-batch Reactor 248
17.5 Dynamics of Ideally Mixed Fermentation Reactor 252
17.6 Linearization of the Model for the Continuous Reactor 254
References 258
18 Physiological Modeling: Glucose-Insulin Dynamics and Cardiovascular Modeling 259
18.1 Introduction to Physiological Models 259
18.2 Modeling of Glucose and Insulin Levels 260
18.3 Steady-state Analysis 262
18.4 Dynamic Analysis 263
18.5 The Bergman Minimal Model 264
18.6 Introduction to Cardiovascular Modeling 264
18.7 Simple Model Using Aorta Compliance and Peripheral Resistance 265
18.8 Modeling Heart Rate Variability using a Baroreflex Model 268
References 271
Trang 1119 Introduction to Black Box Modeling 273
19.1 Need for Different Model Types 273
19.2 Modeling steps 274
19.3 Data Preconditioning 275
19.4 Selection of Independent Model Variables 275
19.5 Model Order Selection 276
19.6 Model Linearity 277
19.7 Model Extrapolation 277
19.8 Model Evaluation 277
20 Basics of Linear Algebra 279
20.1 Introduction 279
20.2 Inner and Outer Product 280
20.3 Special Matrices and Vectors 281
20.4 Gauss–Jordan Elimination, Rank and Singularity 281
20.5 Determinant of a matrix 283
20.6 The Inverse of a Matrix 284
20.7 Inverse of a Singular Matrix 285
20.8 Generalized Least Squares 287
20.9 Eigen Values and Eigen Vectors 288
References 290
21 Data Conditioning 291
21.1 Examining the Data 291
21.2 Detecting and Removing Bad Data 292
21.3 Filling in Missing Data 295
21.4 Scaling of Variables 295
21.5 Identification of Time Lags 296
21.6 Smoothing and Filtering a Signal 297
21.7 Initial Model Structure 302
References 304
22 Principal Component Analysis 305
22.1 Introduction 305
22.2 PCA Decomposition 306
22.3 Explained Variance 308
22.4 PCA Graphical User Interface 309
22.5 Case Study: Demographic data 310
22.6 Case Study: Reactor Data 313
22.7 Modeling Statistics 314
References 316
23 Partial Least Squares 317
23.1 Problem Definition 317
23.2 The PLS Algorithm 318
23.3 Dealing with Non-linearities 319
23.4 Dynamic Extensions of PLS 320
23.5 Modeling Examples 321
References 325
Trang 12Contents [ix]
24 Time-series Identification 327
24.1 Mechanistic Non-linear Models 327
24.2 Empirical (linear) Dynamic Models 327
24.3 The Least Squares Method 328
24.4 Cross-correlation and Autocorrelation 329
24.5 The Prediction Error Method 331
24.6 Identification Examples 332
24.7 Design of Plant Experiments 337
References 340
25 Discrete Linear and Non-linear State Space Modeling 341
25.1 Introduction 341
25.2 State Space Model Identification 342
25.3 Examples of State Space Model Identification 343
References 348
26 Model Reduction 349
26.1 Model Reduction in the Frequency Domain 349
26.2 Transfer Functions in the Frequency Domain 350
26.3 Example of Basic Frequency-weighted Model Reduction 351
26.4 Balancing of Gramians 353
26.5 Examples of Model State Reduction Techniques 356
References 360
27 Neural Networks 361
27.1 The Structure of an Artificial Neural Network 361
27.2 The Training of Artificial Neural Networks 363
27.3 The Standard Back Propagation Algorithm 364
27.4 Recurrent Neural Networks 367
27.5 Neural Network Applications and Issues 370
27.6 Examples of Models 372
References 379
28 Fuzzy Modeling 381
28.1 Mamdani Fuzzy Models 381
28.2 Takagi-Sugeno Fuzzy Models 382
28.3 Modeling Methodology 384
28.4 Example of Fuzzy Modeling 384
28.5 Data Clustering 386
28.6 Non-linear Process Modeling 391
References 397
29 Neuro Fuzzy Modeling 399
29.1 Introduction 399
29.2 Network Architecture 399
29.3 Calculation of Model Parameters 401
29.4 Identification Examples 403
References 410
30 Hybrid Models 413
30.1 Introduction 413
30.2 Methodology 414
30.3 Approaches for Different Process Types 424
30.4 Bioreactor Case Study 436
Literature 438
Trang 1331 Introduction to Process Control and Instrumentation 439
31.1 Introduction 439
31.2 Process Control Goals 440
31.3 The Measuring Device 444
31.4 The Control Device 449
31.5 The Controller 451
31.6 Simulating the Controlled Process 452
References 453
32 Behaviour of Controlled Processes 455
32.1 Purpose of Control 455
32.2 Controller Equations 457
32.3 Frequency Response Analysis of the Process 458
32.4 Frequency Response of Controllers 460
32.5 Controller Tuning Guidelines 462
References 464
33 Design of Control Schemes 465
33.1 Procedure 465
33.2 Example: Desulphurization Process 472
33.3 Optimal Control 475
33.4 Extension of the Control Scheme 478
33.5 Final Considerations 485
34 Control of Distillation Columns 487
34.1 Control Scheme for a Distillation Column 487
34.2 Material and Energy Balance Control 495
Summary 500
References 501
Appendix 34.I Impact of Vapor Flow Variations on Liquid Holdup 501
Appendix 34.II Ratio Control for Liquid and Vapor Flow in the Column 502
35 Control of a Fluid Catalytic Cracker 503
35.1 Introduction 503
35.2 Initial Input–output Variable Selection 505
35.3 Extension of the Basic Control Scheme 509
35.4 Selection of the Final Control Scheme 510
References 514
Appendix A Modeling an Extraction Process 515
A1: Problem Analysis 515
A2: Dynamic Process Model Development 517
A3: Dynamic Process Model Analysis 521
A4: Dynamic Process Simulation 524
A5: Process Control Simulation 530
Hints 534
Index 535
Trang 14[xi]
FOREWORD
In 1970, Brian Roffel and I started an undergraduate course on process dynamics and control, actually the first one for future chemical and material engineers in The Netherlands Our idea was to teach something of general value, so we decided to focus on process modeling Students received a verbal description of a particular chemical or physical process,
to be transformed into a mathematical model To our surprise, students appeared to be highly motivated; they spent much additional time in developing the equations Some of them wanted to do it all by themselves and even refused to benefit from our advice Maybe part of the fun was to be creative, an essential ingredient in model building to complement the systematic approach
In fact, models are situation-dependent Already about 50 years ago we ran into a clear-cut case at Shell, during the development of dynamic models for distillation columns There we faced the problem of defining the response of the column pressure Chemical engineers told
us that this response is very slow: it takes many minutes before the pressure reaches a new equilibrium However, the automation engineers did not agree as in their experience automatic pressure control is relatively fast After some thinking, we discovered that both parties were right: the pressure response can be modeled by a large first-order time constant and a small dead time (representing the sum of smaller time constants) The large time constant dominates the open loop response, while closed loop behavior is limited by the dead time, irrespective of the value of the large time constant Evidently, modeling requires a good insight into the purpose of the model This book provides good guidance for this purpose Most books on process control restrict modeling to control applications However, inside
as well as outside industry many different process models are required, adapted to the specific requirements of the application Consequently there exists a strong need for a comprehensive text about how to model processes in general Fortunately, this excellent book fills in the gap by covering a wide range of methods complemented by a variety of applications It goes all the way from ‘white’ (fundamental) to ‘black’ (empirical) box modeling, including a happy mix in the form of ‘hybrid’ models More specifically, Ben Betlem adapted the systematic approach advocated for software development to modeling in general
Special attention is paid to the influence of the process environment, and to techniques of model simplification The latter can be helpful, among others, in reducing the number of model parameters to be estimated
I wholeheartedly recommend this book, both to students and to professional engineers, and to scientists interested in modeling of processes of any kind
John E Rijnsdorp Emeritus Professor of Process Dynamics and Control
University of Twente The Netherlands
Trang 16[xiii]
PREFACE
Process dynamics and control is an inter-disciplinary area Three disciplines, process, control and information engineering, are of importance, as shown in Fig 1
• Process engineering offers the knowledge about an application
Understanding a process is always the basis of modeling and control A rigorous dynamic process model should be developed to increase the understanding about the operation fundamentals and to test the control hypothesis Experimental model verification is essential to be aware of all uncertainties and peculiarities of the process
• Control engineering offers methods and techniques for (sub-)optimal operation at all
hierarchical control and operational levels
For all process operational problems encountered, an appropriate or promising control method should be tested to meet the defined requirements
• Software engineering offers the means for implementation
The simulation approach or control solution that is developed should be implemented in
an appropriate way and on an appropriate hardware and software platform
control tools
from
control technology
application
from
process technology
information framework
from
information technology
process control
X
Fig 1 Process dynamics and control area in three dimensions
The three disciplines process, control and information technology answer questions such as: for what, why, how, and in which way
Other disciplines are also of interest
• Business management sets the production incentives and defines the coupling between the
production floor and the office
• Human factors study the relation between humans and automation The contents and
format of the supplied information has to meet certain standards to enable the personnel to perform their control and supervisory tasks well This may conflict with the hierarchical structure of the control functions, which will be based on the partitioning of equipment operations For the most part, flexibility of the automation infrastructure can solve these conflicts In addition the degree of automation along the control hierarchy should be chosen with care
Trang 17• Chemical analysis supports quality control The product quality is one of the most
important operation constraints in process operation In this respect, it should be mentioned that quality measurement is often problematic owing to its time delays and its unreliability This can be overcome by a quality estimator based on mathematical principles
This book will create a link between specific applications on the one hand and generalized mathematical methods used for the description of a system on the other
The dynamic systems that will be considered are chemical and physiological systems System behavior will be determined by using analytical mathematical solutions as well as by using simulation, for example Matlab-Simulink Information flow diagrams will be used to reveal the model structure These techniques will enable us to investigate the relationship between system variables and their dependencies
This book is organized in three parts The first part deals with physical modeling, where the model is based on laws of conservation of mass, momentum and energy and additional equations to complete the model description In this case physical insight into the process is necessary It is probably the best model description that can be developed, since this type of model imitates the phenomena that are present in reality However, it can also be a very time-consuming effort
In this first part, numerous unit operations are described and numerous examples have been worked out, to enable the reader to learn by example
The second part of the book deals with empirical modeling Various empirical modeling techniques are used that are all data based Some techniques enable the user to develop linear models; with other techniques non-linear models can be developed It is good practice to always start with the most simple linear model and proceed to more complicated methods only if required
The last part of the book deals with process control Guidelines are given for developing control schemes for entire plants The importance of the process model in controller tuning is shown and control of two process units with multiple inputs and multiple outputs is demonstrated Control becomes increasingly important owing to increased mass and energy integration in process plants In addition, modern plants are highly flexible for the type of feed they can process In modern plants it is also common practice to reduce the size of buffer tanks or eliminate certain buffer tanks altogether Much emphasis is therefore placed
on well-designed and properly operating control systems
Brian Roffel and Ben H.L Betlem
September 2006
Trang 18[xv]
ACKNOWLEDGEMENT
This book makes extensive use of the MATLAB® program, which is distributed by the Mathworks, Inc We are grateful to the Mathworks for permission to include extracts of this code
For MATLAB® product information, please contact:
The MathWorks, Inc
3 Apple Hill Drive
Natick, MA, 01760-2098 USA
For the principal components analysis (PCA) and partial least squares regression (PLS) in Chapters 22 and 23, this book makes use of a PLS Toolbox, which is a product of Eigenvector Research, Inc The PLS Toolbox is a collection of essential and advanced chemometric routines that work within the MATLAB® computational environment We are grateful to Eigenvector for permission For Eigenvector® product information, please contact:
Eigenvector Research, Inc
3905 West Eaglerock Drive
Wenatchee, Wa, 98801 USA
Tel: 509.662.9213
Fax: 509.662.9214
E-mail: bmw@eigenvector.com
Web: www.eigenvector.com
Trang 20Process Dynamics and Control: Modeling for Control and Prediction Brian Roffel and Ben Betlem
© 2006 John Wiley & Sons Ltd
[1]
1 Introduction to Process Modeling
The form and content of dynamic process models are on the one hand determined by the application of the model, and on the other by the available knowledge The application of the model determines the external structure of the model, whereas the available knowledge determines the internal structure Dynamic process models can be used for simulation studies to get information about the process behavior; the models can also be used for control or optimization studies Process knowledge may be available as physical relationships or in the form of process data
This chapter outlines a procedure for developing a mathematical model of a dynamic physical or chemical process, determining the behavior of the system on the basis of the model, and interpreting the results It will show that a systematic method consists of an analysis phase of the original system, a design phase and an evaluation phase Different types of process model are also be reviewed
1.1 Application of Process Models
A model is an image of the reality (a process or system), focused on a predetermined application This image has its limitations, because it is usually based on incomplete knowledge of the system and therefore never represents the complete reality
However, even from an incomplete picture of reality, we may be able to learn several things A model can be tested under extreme circumstances, which is sometimes hard to realize for the true process or system It is, for example, possible to investigate how a chemical plant reacts to disturbances It is also possible to improve the dynamic behavior of
a system, by changing certain design parameters A model should therefore capture the essence of the reality that we like to investigate Is modeling an art or a science? The scientific part is to be able to distinguish what is relevant or not in order to capture the essence
Models are frequently used in science and technology The concept of a model refers to entities varying from mathematical descriptions of a process to a replica of an actual system
A model is seldom a goal in itself It provides always a tool in helping to solve a problem, which benefits from a mathematical description of the system
Applications of models in engineering can be found in (i) Research and Development
This type of model is used for the interpretation of knowledge or measurements An example
is the description of chemical reaction kinetics from a laboratory set-up Models for research purposes should preferably be based on physical principles, since they provide more insight into the coherence as to understand the importance of certain phenomena being observed
Another application of process models is in (ii) Process Design These types of model are
frequently used to design and build (pilot) plants and evaluate safety issues and economical aspects
Models are also used in (iii) Planning and Scheduling These models are often simple
static linear models in which the required plant capacity, product type and quality are the independent model variables
Another important application of models is in (iv) Process Optimization These models
are primarily static physical models although for smaller process plants they could also be dynamic models For debottlenecking purposes, steady-state models will suffice
Trang 21Optimization of the operation of batch processes requires dynamic models Optimization models can often be derived from the design models through appropriate simplification There is, however, a shift in the degrees of freedom from design variables to control variables or process conditions
In process operation, process models are often used for (v) Prediction and Control
Application of models for prediction is useful when it is difficult to measure certain product qualities, such as the properties of polymers, for example the average molecular weight These models find also application in situations where gas chromatographs are used for composition analysis but where the process conditions are extreme or such that the gas chromatograph is prone to failure, for example because of frequent plugging of the sampling system
Process models are also used in process control applications, especially since the development of model-based predictive control These models are usually empirical models, they can not be too complex due to the online application of the models
1.2 Dynamic Systems Modeling
Modeling is the procedure to formulate the dynamic effects of the system that will be considered into mathematical equations The dynamic behavior can be characterized by the dynamic responses of the system to manipulated inputs and disturbances, taking into account the initial conditions of the system
The outputs of the system are the dependent variables that characterize and describe the response of the system The manipulated inputs of the system are the adjustable independent variables that are not influenced by the system The disturbances are external changes that cannot be influenced, they do, however, have an impact on the behavior of the system These changes usually have a random character, such as the environmental temperature, feed or composition changes, etc The initial conditions are values that describe the state of the system at the beginning For batch processes this is often the initial state and for continuous processes it is often the state in the operating point
Manipulated inputs and disturbances cannot always be clearly separated They both have
an impact on the system The former can be adjusted independently; the latter cannot be adjusted This distinction is especially relevant for controlled systems The inputs are then used to compensate the effects of the disturbances, such that the system is kept in a desired state or brought to a desired state
Every book in the area of process modeling gives another definition of the term “model” The definition that will be used here is a combination of the ideas of Eykhoff (1974) and Hangos and Cameron (2003)
A model of a system is:
• a representation of the essential aspects of the system
• in a suitable (mathematical) form
• that can be experimentally verified
• in order to clarify questions about the system
This definition incorporates the goal, the contents (the subject) as well as the form of the model The goal is to find fitting answers to questions about the system The subject for modeling is the representation of the essential aspects of the system The aspects of the system should in principle be verifiable The form of the model is determined by its application Often an initial model that serves as a starting point, is transformed to a desirable form, to be able to make a statement about the behavior of the system
Trang 221 Introduction to Process Modeling 3
At the extremes there are two types of models models that only contain physical-chemical relationships (the so-called “white box” or mechanistic models) and models that are entirely based on experiments (the so-called “black box” or experimental models) In the first category, only the system parameters are measured or known from the literature In that case
it is assumed that the structure of the model representation is entirely correct In the second case, also the relationship(s) are experimentally determined Between the two extremes there
is a grey area In many cases, some parts of the model, especially the balances, are based on physical relationships, whereas other parts are determined experimentally This specifically holds for parameters in relationships, which often have a limited range of validity However, the experimental parts can also refer to entire relationships, such as equations for the rate of reactions in biochemical processes
In the sequel the most important aspects of modeling will be discussed
1 A model is a tool and no goal in itself
The first phase in modeling is a description of the goal This determines the boundaries of the system (which part of the system and the environment should be considered) and the level of detail (to which extent of detail should the system be modeled) The goal should be reasonably clear A well-known rule of thumb is that the problem is already solved for 50% when the problem definition is clearly stated
2 Universal models are uneconomical
It does usually not make sense to develop models that fit several purposes Engineering models could be developed for design, economic studies, operation, control, safety and special cases However, all these goals have different requirements with respect to the level of detail and have different degrees of freedom (design variables versus control variables, etc.)
3 The complexity of the model should be in line with the goal
When modeling, one should try to develop the simplest possible form of the model that is required to achieve the set goal Limitation of the complexity is not only useful from an efficiency point of view, a too comprehensive, often unbalanced model, is undesirable and hides the true process behavior The purpose of the model
is often to provide insight This is only possible if the formulation of the model is limited to the essential details This is not always an easy task, since the essential phenomena are not always clear A useful addition to the modeling steps is a sensitivity analysis, which can give an indication which relationships determine the result of the model
4 Hierarchy in the model
Modeling is related to (experimental) observation It is well known that during observation, the human brain uses hierarchical models A triangle is observed by looking at the individual corners and the entire structure This is also the starting point for modeling The model comprises a minimum of three hierarchical levels: the system, the individual relationships and the parameters Usually a system comprises several subsystems, each with a separate function To understand the system, knowledge is required of the individual parts and their dependencies
5 Level of detail
This is a difficult and important subject Figure 1.1 shows the three dimensions in which the level of detail can be represented: time, space and function Models can encompass a large time frame Then only the large time constant should be considered and the remainder of the system can be described statically If short times are important, small time constants become also important and the long-term effects can be considered as integrators (process output is an integration of the input) This will be discussed in more detail in a later chapter
Trang 23timehorizon
spatialdistribution functional
distribution
Fig 1.1 Level of detail in three dimensions
For the spatial description it is relevant to know whether the system can be considered to be lumped or not Lumped means that all variables are independent of the location An example is a thermometer, which will be discussed in the sequel A good approximation is that the mercury has the same temperature everywhere, independent of the height and the cross sectional location
In case of distributed systems, variables are location dependent Most variables that change in time also change with respect to location Examples are found in process equipment, such as heat exchangers, tubular reactors and distillation columns
The level of detail of a function can vary considerably The functionality of a system can be considered from the molecular level to the user level An example is
a coffee machine At the user level it is only of interest how this system can be operated to get a cup of coffee of the required amount and quality Knowledge about how, how fast and to what extent the coffee aroma is extracted from the powder is not required In order to understand what really happens, knowledge at the molecular level may even be required
Usually there is a dependency between the levels of detail of time, space and functionality If the system is considered over a longer period, the spatial distribution and functionality will require less detail
6 Modeling based on network components versus modeling based on balances
Mechanical, electro-mechanical and flow systems can often be modeled on the basis
of elements (resistance, condenser, induction, transistor) of which the network is composed This is often also possible for thermal systems In many cases these elements can be described by linear relationships because they do not exceed the operating region When combining these thermal systems with chemical systems, this network structure is not so clear anymore The starting point in this case is usually the mass, energy and component balances The balances can often be written as a network of differential equations and ordinary equations But the structure is not recognizable as a network of individual components Forcing such a system into a network of analog electrical components may violate the true situation
7 A model cannot explain more than it contains
Modeling and simulation may enhance the insight, clarify dependencies, predict behavior, explore the system boundaries; however, they will not reveal knowledge that is unknown A model is a reflection of all the experiments that have been performed
Trang 241 Introduction to Process Modeling 5
8 Modeling is a creative process with a certain degree of freedom
The problem statement, the definition of the goal, the process analysis, the design and model analysis are all steps in which choices have to be made Especially important are the assumptions The final result will be dependent on the knowledge and attitude of the model developer
1.3 Modeling Steps
Figure 1.2 shows the steps that are involved in the modeling process in detail There are three main phases: system analysis, model design and model analysis These phases can be further subdivided into smaller steps By using the example of a thermometer, these steps will be clarified Problems during model development are:
• What should be modeled?
• What is the desired level of detail?
• When is the model complete?
• When can a variable be ignored or simplified?
These questions are not all independent The answers to the third and fourth question
depend on the answers to the first two One could state that during the system analysis phase
these questions should be answered The answers are obtained by formulating the goals of the model on the one hand and by considering the system and the environment in which it operates on the other hand This should provide sufficient understanding as to what should be modeled
In the model design phase the real model is developed and when appropriate,
implemented and verified In this phase the first question to be answered should be how the model should look like
The starting point is the design of a basic structure that can be used to realize the goal In case of physically based modeling, this structure is more or less fixed: differential equations with additional algebraic equations
With these models the behavior of a variable in time can be investigated Also other types
of model are possible Examples are so-called experimental models, or black-box models, such as fuzzy models or neural network models The design of these models proceeds using slightly different sub-steps, which will be discussed later
The verification and validation of the implemented model by using data is part of the design phase The boundary between system analysis and design is not always entirely clear When the system analyst investigates the system, he or she often thinks already in terms of modeling During the system analysis phase it is recommended to limit oneself to the analysis of the physical and chemical phenomena that should be taken into account (the
“what”), whereas during the modeling process the way in which these phenomena are accounted for is the key focus (the “how”)
In the model analysis phase the model is used to realize the goals Often the model
behavior is determined through simulation studies, but the model can also be transformed to another form, as a result of which the model behavior can be determined An example is transformation of the model to the frequency domain These types of model give information
on how input signals are transformed to output signals for different frequencies of which the input signal is composed The boundary between model design and model analysis is also not always clear-cut Implementation and transformation of the model are sometimes part of the model design
Trang 25context analysis
context model
laws of conservation
knowledge
system-experiments
function analysis
- justification of assumptions
transformation simulation
specific model
or simulation
information collection
problem definition
test data train data
interpretation evaluation
As mentioned before, the definition phase is the most important phase Feedback does not happen until the evaluation phase Then it will become clear whether the goals are met During the problem analysis phase, the environmental model is developed This is an information flow diagram representing the process inputs and outputs as shown in Fig 1.3 for the case of a simple thermometer
Fig 1.3 Thermometer schematic with environmental diagram
This representation is also called input–output model Using the previously defined goal, the level of detail of the model with respect to time, location and functionality has to be
Trang 261 Introduction to Process Modeling 7 determined The accuracy of a dynamic model is primarily determined by its position in the time hierarchy and the corresponding frequency spectrum The lower the time hierarchy, the higher the frequencies that should be covered The determination of the level of detail cannot
be postponed to the design phase Parallel to the analysis of the system, it is already useful during this phase to investigate in which way the model can be verified
During the system analysis1 one focuses on the system itself, in order to investigate which physical-chemical phenomena take place and are relevant with respect to the modeling goal One method that could be used is to find key mechanisms and key components, for example: evaporation (phase equilibrium), mass diffusion or transport, heat convection, conduction or radiation, and liquid flow The key variables could be, amongst others, temperature, pressure and concentration at a certain location Also in this case, one should take into account the required level of detail, the hierarchy within the model and the required accuracy
The modeling steps as indicated in Fig 1.2 will be illustrated by considering the modeling procedure for a thermometer
Step a: Problem Definition
The starting point for the problem definition is: develop a model for a liquid thermometer as shown in Fig 1.3 This looks like a simple problem, however, it is easy to get lost in details Important questions that could surface are: “how does the thermometer function exactly?” and “what can be ignored?” First the problem definition and the goal should be clearly stated What should be modeled and how?
Examples of problem definitions could be:
• What type of behavior does the height of the mercury show when the environmental temperature changes? This question could be answered qualitatively
• How does the height change exactly? Is this change linearly dependent on the temperature? This is a quantitative question requiring a quantitative answer
• Can the thermometer follow fast changing temperatures? This is also a quantitative question with respect to the system dynamics
• What determines the speed of change? This is a question with respect to relative sensitivities
• Is it possible to use alcohol instead of mercury for certain applications? This is a design question
The goal that should be met in this case will be formulated as:
• The model should provide insight into those factors that determine the speed of change of the indication of the height of the mercury of the thermometer (see Fig 1.4a)
The requirements for the model become then:
• Format The model should be in the form of differential equations with additional
equations, enabling us to perform a simulation This is a very commonly used format
• Level of detail The thermometer will be considered as we view it, hence not at a
molecular level
The most important starting point is the assumption that the temperature in the thermometer does not have a spatial distribution and has the same temperature everywhere
It is assumed that this assumption also holds when the thermometer is not entirely submerged
in the medium of which the temperature should be measured The impact of the volumetric expansion of the glass and the heat transfer through the glass on the operation of the
1 If the system is a process, one could use the term “process analysis”
Trang 27thermometer, have to be further investigated These two factors have an impact on the way in which the mercury level increases or decreases when the temperature changes
Fig 1.4 Detailed (a) and simplified (b) context diagram of the thermometer
Step b: Context Analysis
Let us assume that the model we are interested in is a model that will determine the height of the mercury as a function of the environmental temperature (Fig 1.4b) Therefore, there is only one input and one output, the cause and effect The number of inputs, however, can be disputed, since also the dynamic behavior of level changes will be modeled Other influences from the environment could be:
• Properties of the medium in which the measurement takes place
The heat transfer from the medium to the glass will be determining for the rate of change
of the mercury level Heat transfer will also be determined by the state of aggregation of the medium (gas, liquid or solid), conduction and flow properties of the medium in which the temperature measurement takes place The heat transfer coefficients for non-flowing gas, flowing liquid or boiling liquid are more than a factor of thousand apart In a turbulent air flow, heat is transferred much faster than in a stagnant air flow For a temperature measurement in a gas flow with fast changing temperatures, the velocity of the gas will be an additional model input variable In the sequel it is assumed that the heat transfer conditions are constant
• Radiation to and from the environment
To measure temperatures accurately is difficult, since the measurement is affected by radiation to and from objects with a different temperature, close to the actual measurement This impact is therefore dependent on the situation
• Heat losses through the contact point
The way the thermometer is installed may lead to heat transport through the contact point
(Q bridge)
The result of the context analysis is an input–output model with assumptions about environmental effects that can be ignored In the case of the thermometer it is assumed that only the medium temperature is relevant and that the heat transfer conditions are constant
Step c: Function Analysis
A liquid thermometer works on the principle that the specific mass or density is a function of the temperature The volumetric expansion of the volume is therefore a measure for the increase in temperature
The temperature of the liquid inside the thermometer does not change instantaneously with the environmental temperature Mercury and glass have a heat capacity and increase in temperature as a result of heat conduction through the material
It is our experience that most modeling exercises tend to start with a description of the mercury inside the thermometer (the state of the system) in the form of a differential equation for the density or the volume of the mercury
Trang 281 Introduction to Process Modeling 9 However, the density and the volume change instantaneously with the temperature The density increases when the volume decreases and vice versa, since the mass of the mercury is constant:
0)
(
=+
=
=
dt
d V dt
dV dt
V d dt
The following design activities can be distinguished:
• determine the assumptions
• determine the model structure
• determine the model equations
• determine the model parameters
• model verification
• model validation
Diagrams are an efficient way to portray the model structure The types of diagram that are suitable are ‘data-flow-diagrams’, which show the relationships between functions (the data flows) or function flow diagrams that portray relationships on the basis of variables (the relationships between the data are the functions) In this example, data flow diagrams, also called information flow diagrams will be used They relate closely to the input–output models that were used in a previous phase The functions in the model for the description of the dynamic behavior are differential equations and additional equations These equations contain parameters
The formulation of the assumptions is an activity that proceeds in parallel with the model structuring, the design of the equations and the determination of the parameters The assumptions can be propositions about the way the system functions, they have to be further verified Assumptions can also be simplifications that are made in the derivation of the equations or the determination of the parameters In equations it often happens that one term
is much smaller than another term and subsequently can be ignored An example is the contribution of flow compared to the radiation in a heat balance For parameters it usually concerns the assumption that a parameter value is constant (independent of temperature, pressure and composition) An example is the density or heat capacity which is usually assumed to be constant in a limited operating range
Not all parameters in the model will be known or can be retrieved from data bases, some may have to be determined experimentally This can happen in two different ways: the parameter in question can be measured directly or the system is “trained” using real operating data In the last case the parameters are determined by adjusting the parameters such that the modeling error is minimal The modeling error is the difference between the model output and the corresponding true measurement In both cases an experimental data set of the system is required
Verification and validation of the designed model is an essential step Verification means that the behavior of the model should agree with the behavior of the system under study Validation, however, refers to the absolute values the model produces For dynamic systems
Trang 29it is always necessary to check whether the stationary values over the entire operating region are realistic At this point one should decide whether the assumptions that have been made are correct Even if the model parameters are trained based on a real situation, this check is required However, on the other hand, the danger exists that the parameters only describe the situation for which they were trained It could also happen that the model is overtrained This occurs in situations where there are too many model parameters compared to the number of measurements In that case the measurement errors are also modeled It is obvious that the training data set and test data set should be different
Step d: Model Relationships with Assumptions
The model that follows from the function analysis is shown in Fig 1.5 The temperature of the mercury rises or falls as a result of the change in environmental temperature
The level of the mercury follows from the mercury temperature For the temperature it holds that it rises as long as there is heat transport, in other words as long as there is a temperature difference between the mercury and the environment The heat transport is proportional to the temperature difference
Fig 1.5 Behavioral model of the thermometer
The speed at which the temperature rises is determined by the heat capacity:
mercury glass
glass p mercury
mercury
dt
dT m c m
γ volumetric expansion coefficient, 1/K
The height of the mercury is a function of the temperature according to:
0 0
0
tube tube
A
V h A
V V h
Trang 301 Introduction to Process Modeling 11
To be able to make realistic assumptions, additional knowledge on the physical properties
is required Table 1.1 shows the properties of various materials Ethanol has been added to the Table, since it is an alternative for the use of mercury Water is shown as a reference Table 1.2 shows some guidelines for the heat transfer
Table 1.1 Physical properties at 20 oC and 1 bar
coefficient λ, W/m.K
heat capacity cp, J/kg.K volumetric expansion
coefficient γ, 1/K
density ρ, kg/m3
glass 0.8–1.2 0.8–1.0 × 103 0.1–0.3 × 10–4 2.2–3.0 × 103
Source: David, R.L (ed.) Handbook of Physics and Chemistry, 75th edn: CRC Press, 1994
Table 1.2 Heat transfer coefficient at a fixed surface
situation heat transfer coefficient α,
Source: Bird, R.B Stewart, W.E., and Lightfoot, E.N
Transport Phenomena: Wiley, 1960
The model shown in Eqns (1.2) and (1.3) is based on several assumptions:
• The mercury temperature can be lumped
The mercury inside the thermometer has the same temperature everywhere (lumped) This means that there is no temperature gradient in radial or axial direction This assumption is realistic, since the thermal heat conduction of mercury is relatively large (about 10 times the value for glass) Whether total submersion in the medium makes any difference on the temperature can easily be verified
• Physical properties do not depend on the temperature
In Eqns (1.2) and (1.3) it is assumed that the physical properties c p,mercury and c p,glass and
expansion of mercury over a large range is relatively constant, the difference between
0 °C and 100 °C is only 0.5%
• Volumetric expansion of the glass can be accounted for
As a result of the volumetric expansion of the glass, the surface area and the volume increase Owing to the increased surface area for heat transfer, the heat exchange increases However, it can be calculated that this effect is very small The increase of the volume, however, has a significant effect on the mercury level The volumetric expansion coefficient of mercury is 1.8 × 10–4 K–1, the value for glass is in the order of 0.1–0.3 × 10–4 K–1.2 By ignoring the volumetric expansion of glass, an error is made in the order of 10% A possible solution is to use a modified value for mercury, which is the difference between the two expansion coefficients
2 For glass usually the linear expansion coefficient α is given, since it is a solid When the expansion is the same in all directions, γ = 3α
Trang 31Fig 1.6 Volumetric expansion coefficients for various liquids
• Models for heat transfer
In the proposed model it is implicitly assumed that the temperatures of the glass and the mercury are the same It is, however, questionable whether this is true The temperature of the glass changes always before the mercury temperature, because the glass is exposed to the medium This effect could be taken into account if required
In a stationary situation, the thermal resistance (inverse of the heat transfer coefficient) can be written as:
mercury glass glass glass glass env
d
11
1
αλ
in which:
U overall heat transfer coefficient, J/m2.K
α heat transfer coefficient, J/m2.K
d thickness, m
λ heat conduction coefficient, J/m.K
There are three resistances in series: the resistance for heat transfer from the environment to the glass, the resistance for heat transfer through the glass and the resistance for heat transfer from the glass to the mercury The magnitude of these three individual terms will indicate whether the assumption that the mercury temperature is equal to the glass temperature is justified Generally, the resistance between the glass and mercury will be small (αglas,mercury > 200 J/m2.K), the heat conduction term also indicates a small resistance (λglass/dglass = 103 J/m2.K), but the resistance between the glass and the medium can vary considerably, depending on the turbulence in the medium (10 < αenv,glass < 104 J/m2.K)
The so-called Biot number NB indicates to what extent convection or conduction is the limiting factor in the heat transport NB is the dimensionless ratio between the
conductive and convective resistance:
Trang 321 Introduction to Process Modeling 13
λ
αd
When the Biot number is small (NB < 0.1), then the resistance of the glass is small
(relatively large conduction) and the glass will therefore have a uniform temperature Two possibilities that often happen in practice will be considered:
a Heat transfer between the environment and the thermometer is rate limiting
If the thermometer measures the air temperature, the heat transfer from the environment to the glass will be the limiting factor for the rate at which the mercury level changes The value for the heat transfer coefficient follows from Table 1.1:
01.01
d N
b Heat transfer between the environment and the thermometer is not rate limiting
In almost all other cases, the heat conduction through the glass together with the heat transfer from the glass to the mercury will be the limiting factor for heat transfer Assume that the thermometer is submerged in a boiling liquid Then the heat transfer coefficient is in the order of αenv,glass = 1000 W/m2.K and NB is a factor of one hundred
larger It is also not allowed to assume that the glass and mercury temperature are the same For the glass and mercury temperature separate differential equations should be used The heat transfer through the glass can then be modeled by several differential equations in series
• Uniform mercury tube
In the model it is assumed that the cross sectional area of the mercury tube does not depend on the height This should be verified since it will depend on the construction
of the tube
Often, as is the case here, the description of the assumptions will be more elaborate than the mathematical formulation of the model The most uncertain assumptions will have to be verified In this example this is the assumption that the temperature of the mercury and the glass are the same
Step e: Verification and Validation
In this step it should be checked whether the most important assumptions are realistic A simple experiment is to submerge the thermometer in a bath of warm water and wait until an equilibrium situation is obtained Subsequently place the thermometer is a bath with water at room temperature and measure the change of temperature with time (the dynamic behavior) and compare it to the model results The experiment should be repeated by placing the thermometer from the warm bath in air at room temperature
Trang 33This simple experiment will enable us to verify whether the assumption we made about the heat transfer coefficient is correct
Another important assumption is that the temperature of the glass and mercury are the same If these temperatures differ, the temperature of the glass will change first and subsequently the mercury temperature will change A consequence is that for a temperature increase, first the glass would expand and the mercury level would fall, but the longer-term effect would be that the mercury level would rise Whether this happens in reality is easy to check
1.3.3 Model Analysis
The developed model is the starting point for a model analysis or analysis of the behavior of the thermometer The model will have to be solved mathematically, simulated by using a computer or be transformed to a mathematical form that is suitable for the analysis Examples of such a suitable form are, for example, transfer functions and state models Transfer functions are linearized input–output relationships This form of the model will show how the output signal changes for a given change in input signal State models are differential equations in matrix format This form is, amongst others, suitable to perform a stability analysis
A first step in determining the model behavior could be a sensitivity analysis It is then investigated how the model outputs change when the model inputs or parameters change In addition it can be determined which relationships are relevant or can be omitted With this information the model can be further simplified or extended and one can ensure that it only captures the essence
Step f: Transformation to a Simulation Model
The model consists of two equations:
mercury env mercury glass
glass glass p mercury mercury
p
T T dt
dT UA
m c m c
−
=
+ , ,
(1.8) and
The drawback of this model is that several values are difficult to determine, such as: mmercury ,
m glass , A glass , U, V 0 , A tube and γmercury,net The overall heat transfer coefficient U determines the dynamic behavior This includes the heat transfer from the environment to the glass and the heat conduction through the glass Both effects are difficult to determine accurately For the
static behavior V0 and Atube are important An error of 10% in their values results in an error
of 10% in the model Therefore the model should be transferred to a form that is suitable for
and K Equation (1.8) can then be transformed to:
mercury env
UA
m c m
=
Trang 341 Introduction to Process Modeling 15
τ is a characteristic time, called time constant It indicates how fast the mercury and glass
experimentally verifiable quantity and can be estimated using Eqn (1.11) From Table 1.1 it
is clear that only the heat capacity of glass is relevant With estimated values for the volume and cross sectional area on the basis of a glass tube of 10 cm long, a diameter of 2 mm and a wall thickness of 1 mm, the following values are obtained:
Table 1.3. Values of thermometer parameters
M glass =V glass × ρglass = 0.5 ml × 2 × 103 kg/m3 = 1 × 10–3 kg
Two extreme cases can be considered:
• The heat transfer between the air and the thermometer is rate limiting The value for the overall heat transfer coefficient follows from Table 1.2 and is U = 10 W/m2.K Using the values before, this results in an estimated value of τ = 200 s
• Heat transfer through the glass is rate limiting In this case the temperature of the glass
and mercury will not be the same and the glass temperature will show a gradient The
1 W/m.K/10–3 m = 103 W/m2.K Consequently, the estimated time constant τ = 2 s
From experiments it can be shown that these values are realistic In the second case in which the heat conduction is rate limiting, a more complicated model will have to be developed, because temperature changes are fast and more model detail is required to capture this If the thermometer is part of a larger system in which changes on a time scale of minutes are important, dynamic thermometer temperature changes are relevant in the first case, whereas in the second case these changes can be assumed to be instantaneous
Equation (1.9) can be written as:
T K
with
net mercury tubeA
V
K indicates how the height increases with temperature This is reflected in the thermometer
scale Many errors can be accounted for by proper scaling, such as errors in the assumption that the physical properties are constant The scaling can be determined by calibration For a
calibration thermometer, K is a function of temperature, since small corrections are required
for the expansion of mercury and glass It is then essential that the thermometer is submerged
in the medium
The global model is more accurate than the detailed model, since the constants of the global model can be experimentally determined, whereas this is not possible for the detailed model
Trang 35Step g: Interpretation and Evaluation
Upon a step change in the temperature from 20 °C to 50 °C, the height of the mercury in the thermometer will slowly rise as shown in Fig 1.7
Fig 1.7 Thermometer response to a step change in environmental temperature
The dynamic behavior is in agreement with the interpretation from the simple behavioral model as depicted in Fig 1.5 The thermometer behaves like a first-order system with a mercury height which varies linearly with the temperature
The speed at which this will take place depends on the heat transfer coefficient from the environment to the glass In air the temperature change is gradual, in boiling water the temperature rise is almost instantaneous Calibration thermometers always use mercury as a liquid The accuracy is approximately 0.1 °C The thermometer should always be fully submerged, the scale of these thermometers is not linear
• specific model of simulation
Diagrams are a good means to illustrate these deliverables: the environmental model is illustrated by a context diagram showing the connections with the environment, the behavioral model is supported by data-flow diagrams showing the functions and internal structure and also for simulations often graphical tools are used
The utilization of graphical techniques is not new Context diagrams in the field of process control are usually called input–output diagrams Signal-flow diagrams, electric analogies or bond graphs have been applied in process dynamics before to clarify structures Some simulation packages use block diagrams to input the models into the software The method used in the sequel is “borrowed” from the field of informatics (Yourdon, 1989) The representation of the functional structure is done by using so-called data flow diagrams (DFDs, Fig 1.8) There is an essential difference with signal flow diagrams (van der Grinten, 1970) In the latter type of diagrams, first a list is made of all variables and
Trang 361 Introduction to Process Modeling 17
subsequently the relationships are identified (Rademaker, 1971) In developing the DFDs, the reverse order is followed
Fig 1.8 Examples of balances; left: integrator, middle: first-order with external feedback, right:
first-order with internal feedback
A DFD consists of functions represented by circles, interconnected by arrows representing the relationships by which the variables are transported The sign next to the arrow indicates whether the variable has a positive or negative impact on a function DFDs can be hierarchically structured Every function can be further detailed into sub-DFDs At the lowest level, every function is defined by its equation Differential equations are represented
by a buffer, because they have a “status memory” The new state is determined by the present state and a new input or external influence At start-up, this memory should have an initial value If the state only occurs in the balance in the derivative (left-hand side of balance), then a double arrow is used (Fig 1.8, left) and if a state also appears as a state feedback (right-hand side of balance), then two single arrows are used (Fig 1.8, right) By this method distinction can be made between balances with and without feedback
The graphical representation ensures that the structure of the model is shown and clarified
It also ensures that missing relationships are shown and which relationships have to be worked out in more detail The behavioral model has the same inputs and outputs as the environmental model Throughout the book these diagrams will be used
1.4.2 Interpretation of Behavioral Diagrams
During the model design phase, but also during the model analysis phase, diagrams are useful The diagrams consist of linked balances and additional equations The behavior can be derived from the way the balances are serially interconnected in chains and linked in parallel into loops The chains will correspond to convective and conductive behavior Every loop can be associated with a closed loop gain They generate the so-called poles of the system In addition, parallel influences can act on a chain or loop They generate the so-called zeros of the system Convective, conductive, closed-loop and parallel process behavior will be discussed in a later chapter
data-flow-Systems can be arranged, depending on their DFD-structure into several categories:
• balance without feedback (Fig 1.8, left)
The differential equation has the following form, such as:
out
Q dt
dx
The process concerns a pure integrator An example of this system is a tank or surge drum
Trang 37• Balance with feedback (Fig 1.8, middle and 1.8, right)
In this case, there is a feedback, meaning that the overall differential equation becomes:
)
(x
f Q dt
dx
The feedback is usually negative There are numerous examples of this type of system; one example was the energy balance for the thermometer, Eqn (1.10) This type of system is called a first-order system
• A chain of first-order processes without feedback (Fig 1.9, top)
The equation for one process section of the chain, has a form such as:
) ( ) ( )
• A chain of interactive first-order processes (Fig 1.9, bottom)
The equation for one process section of the chain, has the form:
) , ( ) ,
• Loops of interactive balances (Fig 1.10, left)
Loops are formed by mutual interaction of balances As will be shown in a subsequent chapter, the solution can have several forms The set of equations for two interacting balances, could have the form:
( ) ( )
( ) y g x f Q dt
dy C
y g x f Q dt
dx C
y y
in
x x in
− +
2 , 2
1 , 1
(1.18)
Examples are interacting mass and energy balance (evaporator) or interacting component and energy balance (reactor)
• Parallel influencing of chain or loop (Fig 1.10, right)
Inputs can act parallel to different balances When the influences are opposite, it may lead
to an inverse system response The set of equations for two interacting balances with a
parallel influence from an input variable Qin,1, could have a form such as:
( ) ( ) y g x f Q dt
dy C
y g Q Q dt
dx C
y y
in
x in in
− +
1 , 2
2 , 1 , 1
(1.19)
Trang 381 Introduction to Process Modeling 19
balance
n
Q
=f x{ }n n+
n
Q
=f x{n+1 n+1}+
Fig 1.9 Chain of process sections with internal feedback per section; top: only feed-forward
linking, bottom: also feedback linking
x balance
y balance +
x
Q in,2
y y
Q in,1
x
x balance
y balance +
_
x
y y
Trang 391.5 Types of Model
The different applications of models and the different modeling goals have lead to many different model structures Since models are used as a basis for further decisions, the knowledge should be presented in a usable form The model should not be too complex, nevertheless it should give a sufficiently accurate description of the system The following classifications can be made:
The model will always be a combination of the characteristics mentioned above which will be posed by the requirements which the application poses on the one hand, and the process knowledge and available data on the other hand
1.5.1 White Box versus Black Box Models
White box models are based on physical and chemical laws of conservation, such as mass
balance, component balance, momentum balance and energy balance They are also called
first principles or mechanistic models The models give physical insight into the process and
explain the process behavior in terms of state variables and measured variables The state variable of the model is the variable whose rate of change is described by the conservation balance These models can already be developed when the process does not yet exist The dynamic equations are supplemented with algebraic equations describing heat and mass transfer, kinetics, etc Developing these models is very time consuming
Black box models or empirical models do not describe the physical phenomena of the
process, they are based on input/output data and only describe the relationship between the measured input and output data of the process These models are useful when limited time is available for model development and/or when there is insufficient physical understanding of the process Mathematical representations include time series models (such as ARMA, ARX, Box and Jenkins models, recurrent neural network models, recurrent fuzzy models)
It could be that much physical insight is available, but that certain information or understanding is lacking In those cases, physical models could be combined with black box
models; the resulting models are called gray box or hybrid models The latter type of models
will be discussed in chapter 30 Examples are physical models combined with neural network or fuzzy logic models (van Lith, 2002)
In the process industries large amounts of data are available In most modern industries, process control systems are connected to systems that collect process data on a regular basis (e.g 1 minute) and by using special data compression techniques, data can be stored for years Generally, it is difficult to discover dependencies amongst the data, even for an experienced engineer
Especially in an operating environment where quality has become more important than quantity, there is a strong desire to develop input–output models that can be used in advanced control applications, in order to develop control strategies for quality improvement These models are usually discrete linear transfer function (difference equation) type models, which provide a representation of the dynamic behaviour of the process at discrete sampling
Trang 401 Introduction to Process Modeling 21
intervals They are in general much simpler to develop than theoretical models, both their structure and parameters can empirically be identified from plant data
1.5.2 Parametric versus Non-parametric Models
For the large group of dynamic lumped models, which are used for optimization and control, also a classification between parametric and non-parametric models can be made This classification resembles the classification in white-box and black-box models The latter distinction is based on the difference in knowledge content, whereas the former distinction refers to the form of the set of equations
Parametric models are more or less white box or first principle models They consist of a set of equations that express a set of quantities as explicit functions of several independent variables, known as “parameters” Parametric models need exact information about the inner structure and have a limited number of parameters For instance, for the description of the dynamics, the order of the system should be known Therefore, for these models, process knowledge is required Examples are state space models and (pulse) transfer functions Non-parametric models have many parameters and need little information about the inner structure For instance, for the dynamics, only the relevant time horizon should be known
By their structure, they are predictive by nature These models are black box and can be constructed simply from experimental data Examples are step and pulse response functions Structural considerations concern the order (parametric models) or the time-horizon (non-parametric models) of the model Usually, the model is identified by minimizing the error between data and model This can lead to over-modeling by including noise or eventualities The common strategy is to use separate data sets for model identification and for testing Statistical tests can be applied to test the parameter significance
1.5.3 Linear versus Non-linear Models
Linear models have a property that is called superposition If the model is of the form y=f(u),
then the property of superposition states that:
)(
)()
used as an approximation of the true relationship between the process input and process output Linear models often provide an accurate description of reality provided the operating range is limited In that case, a first-order Taylor series approximation is appropriate, where the first-order derivative is used to describe the behavior around the operating point Linear models are much easier to handle mathematically and easier to interpret as the relationship between input and output is explicit
It is generally not easy to solve non-linear models analytically, but sophisticated numerical methods embedded in commercial software packages are available to deal with several classes of non-linear models
Sometimes, linear techniques can be used to describe non-linear process behavior An example is a fuzzy model, discussed in chapter 28, which is a combination of local linear models in distinct operating areas Developing a non-linear model requires much insight and understanding of the developer as to what mechanism underlies the observed data Application of empirical techniques for modeling non-linear process behavior has therefore become very popular, such as the application of neural networks, described in chapter 27